Unit 3: Vector Calculus
Vector Calculus is built around the vector differential operator, del (∇).
In Cartesian coordinates: ∇ = î (∂/∂x) + ĵ (∂/∂y) + k̂ (∂/∂z)
This operator is not a vector itself, but it acts on scalar or vector fields to produce new fields.
Vector Differentiation
This refers to differentiating a vector function of a single scalar variable, like time (t).
If r(t) = x(t)î + y(t)ĵ + z(t)k̂ is the position vector of a particle:
- Velocity (v): v(t) = dr/dt = (dx/dt)î + (dy/dt)ĵ + (dz/dt)k̂
- Acceleration (a): a(t) = dv/dt = d²r/dt² = (d²x/dt²)î + (d²y/dt²)ĵ + (d²z/dt²)k̂
Differentiation Rules: (f is a scalar, A and B are vectors)
- d/dt (A + B) = dA/dt + dB/dt
- d/dt (f A) = (df/dt)A + f(dA/dt) (Product rule)
- d/dt (A · B) = (dA/dt) · B + A · (dB/dt) (Product rule for dot product)
- d/dt (A × B) = (dA/dt) × B + A × (dB/dt) (Product rule for cross product - order matters!)
Directional Derivatives and Normal Derivatives
Directional Derivative
The directional derivative measures the rate of change of a scalar field f(x,y,z) at a specific point P, in a specific direction.
Let the direction be given by a unit vector û.
The directional derivative of f in the direction û is:
Dûf = (∇f) · û
It is the dot product of the gradient of f (see below) and the unit vector of the direction.
- Step 1: Find the gradient, ∇f.
- Step 2: Find the unit vector û. (If given a vector V, û = V / |V|).
- Step 3: Calculate the dot product ∇f · û.
- Step 4: Evaluate the result at the given point P.
Normal Derivative
The "normal derivative" is simply the directional derivative in the direction normal (perpendicular) to a surface. As we will see, the gradient (∇f) is *already* normal to the level surfaces of f. Therefore, the normal derivative is often used to mean the maximum rate of change, which is just the magnitude of the gradient, |∇f|.
Gradient of a Scalar Field (∇f)
The gradient acts on a scalar field (f) and produces a vector field.
grad(f) = ∇f = (∂f/∂x)î + (∂f/∂y)ĵ + (∂f/∂z)k̂
Geometrical Interpretation of Gradient
The gradient ∇f at a point P has a very important physical meaning:
- Direction: ∇f points in the direction of the steepest ascent (maximum rate of change) of the scalar field f.
- Magnitude: |∇f| is the value of this maximum rate of change.
- Normality: ∇f is always normal (perpendicular) to the level surface (or contour line) f(x,y,z) = constant that passes through the point P.
Physical Applications of Gradient:
- Force from Potential: A conservative force F is the negative gradient of its scalar potential energy U.
F = -∇U
- Electric Field from Potential: The electric field E is the negative gradient of the electric potential V.
E = -∇V
Divergence of a Vector Field (∇ · V)
The divergence acts on a vector field (V) and produces a scalar field. It is the dot product of ∇ and V.
If V = Vxî + Vyĵ + Vzk̂:
div(V) = ∇ · V = (∂Vx/∂x) + (∂Vy/∂y) + (∂Vz/∂z)
Physical Interpretation of Divergence
Divergence measures the "outflow" or "source strength" of a vector field from an infinitesimal volume around a point.
- ∇ · V > 0: The point is a source. More field lines "diverge" from this point than enter it. (e.g., a positive charge for an E-field).
- ∇ · V < 0: The point is a sink. More field lines enter this point than leave. (e.g., a negative charge).
- ∇ · V = 0: The field is solenoidal (or "divergence-free"). There are no sources or sinks. The same amount of "flow" enters any volume as leaves it.
Key Application (Maxwell's Equations):
- Gauss's Law: ∇ · E = ρ / ε₀ (Charge density ρ is the source of E).
- Gauss's Law for Magnetism: ∇ · B = 0 (There are no magnetic monopoles, so B-fields are always solenoidal).
Curl of a Vector Field (∇ × V)
The curl acts on a vector field (V) and produces another vector field. It is the cross product of ∇ and V.
It is calculated using a determinant:
| î ĵ k̂ |
curl(V) = ∇ × V = | ∂/∂x ∂/∂y ∂/∂z |
| Vx Vy Vz |
= (∂Vz/∂y - ∂Vy/∂z)î - (∂Vz/∂x - ∂Vx/∂z)ĵ + (∂Vy/∂x - ∂Vx/∂y)k̂
Physical Interpretation of Curl
Curl measures the microscopic "circulation" or "vorticity" of a vector field at a point. Imagine placing a tiny paddlewheel in the field.
- ∇ × V ≠ 0: The field is rotational. The paddlewheel would spin. The vector ∇ × V points along the axis of this rotation (by the right-hand rule). (e.g., water in a whirlpool, a magnetic field around a current-carrying wire).
- ∇ × V = 0: The field is irrotational (or "curl-free"). The paddlewheel would not spin.
Irrotational Fields are Conservative:
A vector field V is conservative if its line integral depends only on the endpoints, not the path.
Key Theorem: A field V is conservative if and only if ∇ × V = 0.
If a field is conservative, it can *always* be written as the gradient of some scalar potential: V = ∇f (or F = -∇U, E = -∇V).
Vector Identities
These are "second-order" derivatives that are fundamental in physics.
- Curl of Gradient is Zero: ∇ × (∇f) = 0
This is always true. The gradient of a scalar field (∇f) is always an irrotational (conservative) vector field.
- Divergence of Curl is Zero: ∇ · (∇ × V) = 0
This is also always true. The curl of a vector field (∇ × V) is always a solenoidal (divergence-free) vector field. (This is why ∇ · B = 0, because B can be written as the curl of the vector potential A, B = ∇ × A).
- Divergence of Gradient (Laplacian): ∇ · (∇f) = ∇²f
This is called the Laplacian of the scalar field f.
∇²f = (∂²f/∂x²) + (∂²f/∂y²) + (∂²f/∂z²)
It appears in many key physics equations (Wave equation, Schrödinger's equation, Poisson's equation ∇²V = -ρ/ε₀).
Vector Integration: Line, Surface, and Volume Integrals
Line Integrals
A line integral sums up contributions along a path (a curve) C. A common form is:
W = ∫C V · dr
- dr = dx î + dy ĵ + dz k̂ (an infinitesimal displacement vector along the path).
- Physical Meaning: This calculates the work done by the vector field V (e.g., a force field F) on a particle moving along the path C.
- If C is a closed path (∮): The line integral ∮ V · dr is called the Circulation of V around C.
Surface Integrals
A surface integral sums up contributions over a 2D surface S. A common form is:
Flux = ∫∫S V · dS
- dS = n̂ dA (an infinitesimal area element dA with a direction n̂ normal to the surface).
- Physical Meaning: This calculates the flux of the vector field V (e.g., an E-field or fluid velocity) flowing through the surface S.
Volume Integrals
A volume integral sums up contributions over a 3D volume V.
Total Q = ∫∫∫V f dV
- dV = dx dy dz (an infinitesimal volume element).
- Physical Meaning: This is used to find total quantities. If f is a scalar field like charge density (ρ), the integral gives the total charge Q inside the volume V.
Gauss's Divergence Theorem
The Divergence Theorem is a powerful tool that connects a volume integral to a closed surface integral. It relates the "sources" inside a volume to the "flux" coming out of its boundary.
Let V be a 3D volume, and let S be the closed surface that bounds V.
∫∫∫V (∇ · V) dV = ∮∮S V · dS
In words:
"The volume integral of the divergence of a vector field V within a volume is equal to the net flux of V through the closed surface that encloses that volume."
Key Application (Deriving Gauss's Law):
We know `Q_enclosed = ∫∫∫ ρ dV`. From Maxwell's equation, `ρ = ε₀(∇ · E)`.
So, `Q_enclosed = ∫∫∫ ε₀(∇ · E) dV = ε₀ ∫∫∫ (∇ · E) dV`.
By the Divergence Theorem, `∫∫∫ (∇ · E) dV = ∮∮ E · dS`.
Therefore, `Q_enclosed = ε₀ (∮∮ E · dS)`, which gives the familiar integral form: ∮∮ E · dS = Qenclosed / ε₀.
Stokes' Theorem
Stokes' Theorem is another powerful tool that connects a surface integral to a closed line integral. It relates the "circulation" over a surface to the "flow" around its boundary.
Let S be an open surface, and let C be the closed curve (loop) that forms the boundary of S.
∫∫S (∇ × V) · dS = ∮C V · dr
In words:
"The surface integral of the curl of a vector field V over a surface is equal to the line integral (circulation) of V around the closed boundary of that surface."
Key Application (Deriving Ampere's Law):
From Maxwell's equation, `∇ × B = μ₀J` (in steady state).
Integrate over a surface S: `∫∫ (∇ × B) · dS = ∫∫ (μ₀J) · dS`.
The right side is `μ₀` times the current Ienclosed passing through the surface.
By Stokes' Theorem, the left side is `∮ B · dr`.
Therefore: ∮ B · dr = μ₀ Ienclosed.